{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:regular-gabor-frame-existence",
  "title": "The Existence Problem for Regular Gabor Frames",
  "canonical_statement": "Does every lattice of density above one admit a Gabor frame with a nice window? No. For every dimension $d > 1$ there are explicit criteria on lattices $\\Lambda \\subset \\mathbb{R}^{2d}$ with $D(\\Lambda) > 1$ such that no function with continuous Zak transform generates a Gabor frame along $\\Lambda$, which answers the existence problem negatively for Schwartz-class windows.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:regular-gabor-frame-existence:event",
      "problem_id": "vibemathed:regular-gabor-frame-existence",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-06-24T00:00:00.000Z",
      "title": "arXiv:2606.26052 - On the existence problem of regular Gabor frames",
      "summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:regular-gabor-frame-existence:attempt"
      ],
      "method_family_ids": [
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      ],
      "source_assertion_ids": [
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    }
  ],
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    {
      "id": "vibemathed:regular-gabor-frame-existence:attempt",
      "problem_id": "vibemathed:regular-gabor-frame-existence",
      "model": "GPT-5.4, Claude Opus 4.7",
      "provider": "OpenAI / Anthropic",
      "attempted_at": "2026-06-24T00:00:00.000Z",
      "human_collaborators": [
        "Jaume de Dios Pont",
        "Lukas Liehr",
        "Mitchell A. Taylor"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The disclosure separates the two roles: GPT-5.4 was used mainly for mathematical exploration, including exploring whether homology-theoretic methods could give a common-zero criterion for two quasiperiodic functions, while Claude Opus 4.7 assisted with the Lean formalization. The authors verified everything independently. The same group's Cantor Fourier frame paper is also in this catalog.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Same group's Cantor Fourier frame result, arXiv:2607.08656",
          "url": "https://arxiv.org/abs/2607.08656",
          "kind": "paper",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "arXiv:2606.26052 - On the existence problem of regular Gabor frames",
          "url": "https://arxiv.org/abs/2606.26052",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Existence Problem for Regular Gabor Frames",
          "url": "https://vibemathed.com/problem/regular-gabor-frame-existence",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:regular-gabor-frame-existence:verification",
      "problem_id": "vibemathed:regular-gabor-frame-existence",
      "solution_event_id": "vibemathed:regular-gabor-frame-existence:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A Lean formalization accompanies the paper; we have not compiled it. arXiv preprint, not peer-reviewed.",
      "sources": [
        {
          "label": "VibeMathed: The Existence Problem for Regular Gabor Frames",
          "url": "https://vibemathed.com/problem/regular-gabor-frame-existence",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2606.26052 - On the existence problem of regular Gabor frames",
          "url": "https://arxiv.org/abs/2606.26052",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2606.26052 - On the existence problem of regular Gabor frames",
      "url": "https://arxiv.org/abs/2606.26052",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Existence Problem for Regular Gabor Frames",
      "url": "https://vibemathed.com/problem/regular-gabor-frame-existence",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:regular-gabor-frame-existence:signal:watch_only",
      "problem_id": "vibemathed:regular-gabor-frame-existence",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}